1. Field of the Invention
The invention is related to the field of electrical resistivity well logging of wellbores drilled through earth formations. More specifically, the invention is related to methods for enhancing the axial resolution of resistivity measurements which have relatively great lateral depth of investigation into the earth formations.
2. Description of the Related Art
Electrical resistivity well logging is used for determining certain properties of earth formations penetrated by wellbores. A typical electrical resistivity measuring instrument is an electromagnetic induction "array" well logging instrument such as described in U.S. Pat. No. 5,452,761 issued to Beard et al. The induction logging instrument described in the Beard et al '761 patent includes a number of receiver coils spaced at various axial distances from a transmitter coil. Alternating current is passed through the transmitter coil, which induces alternating electromagnetic fields in the earth formations. Voltages are induced in the receiver coils as a result of electromagnetic induction phenomena related to the alternating electromagnetic fields. Generally speaking, voltages induced in the axially more distal receiver coils are the result of electromagnetic induction phenomena occurring at greater lateral distances from the axis of the instrument, and the voltages induced in the axially proximal receiver coils are the result of induction phenomena laterally more proximal to the instrument axis. Conversely, the axial sensitivity to the electromagnetic phenomena of the measurements made by the axially proximal receiver coils is finer than the axial sensitivity of the measurements made by the axially more distal receiver coils.
An objective of resistivity well logging is to be able to determine the resistivity of the earth formation at relatively great lateral distances from the instrument axis, where it is less likely that the native fluids in the pore spaces of the formations have been disturbed by infiltration of fluid from the wellbore. This objective, however, is in direct conflict with another objective of resistivity well logging which is to determine resistivity with the greatest practical axial resolution, because many earth formations include thin layers having widely different resistivities. The conflict in these objectives results from the previously described relationship between the spacing of the receiver coils from the transmitter coil and the sensitivity of the receiver signals to the position of the electromagnetic phenomena.
One type of method known in the art for improving the axial resolution of the measurements made by the axially more distal receiver coils is called deconvolution. Generally speaking, deconvolution of resistivity well logs is a process which boosts high spatial frequency components of the measurements made by the axially distal receiver coils. See for example, Looyestijn, W. J., Deconvolution of Petrophysical Logs: Applications and Limitations, Paper W, 23rd Annual Well Logging Symposium, Society of Professional Well Log Analysts (1982). Deconvolution has proven to be unreliable because the processes typically amplify noise in the measurements, and can be numerically unstable.
Another type of method for improving axial resolution of induction logging measurements is known as "enhanced resolution processing". See for example, Nelson R. J. et al, Improved Vertical Resolution of Well Logs by Resolution Matching, Paper JJ, 31st Annual Well Logging Symposium, Society of Professional Well Log Analysts (1990), and Barber, T. et al, Induction Vertical Resolution Enhancement, Physics and Limitations, Paper O, 29th Annual Well Logging Symposium, Society of Professional Well Log Analysts (1988). Generally speaking, enhanced resolution processing uses higher spatial frequency measurement components present in measurements made by receiver coils axially proximal to the transmitter coil and adds the higher spatial frequency components to measurements made by the axially more distal receiver coils. Expressed mathematically: ##EQU1## where the enhanced axial resolution measurement is represented by C.sub.l.sup.eh. The lower and higher axial resolution measurements (those made by the axially proximal and distal receiver coils) are represented, respectively, by C.sub.l and C.sub.h, and C.sub.h.sup.lpf represents the higher axial resolution measurement after low pass filtering. The low pass filter applied to the higher axial resolution measurement, C.sub.h, is selected so that the filtered measurement, C.sub.h.sup.lpf, has the same axial resolution as the low resolution measurement, C.sub.l. The difference quantity in equation (1) therefore represents the higher resolution (higher spatial frequency) information which is absent from the low axial resolution measurements. The filter applied to the higher resolution measurements does not have any effect on the lateral depth of investigation of the measurements, however. The resolution enhancement performed by adding high spatial frequency components from the higher axial resolution measurements to the low axial resolution measurements must necessarily assume that there is substantially no lateral variation in the resistivity of the earth formations. In earth formations of commercial interest, it is typically the case that the resistivity is not laterally constant. Infiltration of fluid from the wellbore to the pore spaces of the formation is the rule rather than the exception, so lateral variations in resistivity can be expected. The enhanced resolution processing methods known in the art do not properly account for this situation.
More recently, induction well logging instruments have been introduced which include a plurality of receivers at various axial distances from a transmitter. The previously described resolution enhancement techniques when applied to these instruments are generalized into so-called "two-dimensional focusing schemes" as described in, Barber, T. A., and Rosthal, R. A., Using a Multiarray induction Tool to Achieve High-Resolution Logs with Minimum Environmental Effects, paper no. 22725 presented at the 1991 SPE Annual Technical Conference and Exhibition, Dallas, Tex., or Zhou, Q., Beard. D., and Tabarovsky, L. A., Numerical Focusing of Induction Logging Measurements, 12th Workshop in Electromagnetic Induction in Earth, International Union of Geodesy and Geophysics (1994). These two dimensional focusing schemes can be expressed mathematically as: ##EQU2## where N represents the number of measurement channels (receiver coils), .sigma..sub.log represents a focused log with a desired lateral depth of investigation and axial resolution, .sigma..sub.a.sup.n represents an unfocused measurement from the n-th measurement channel (receiver coil), and w.sub.n represents a set of filter coefficients to be applied to the n-th measurement channel (receiver coil). These deconvolution and enhanced resolution techniques are, in fact, special cases of a general 2-dimensional focusing scheme, where N=1 and 2, respectively. It can also be shown that this 2-dimensional focusing scheme is based on the assumption of no lateral variation in the resistivity of the earth formation. None of these resolution enhancement schemes properly accounts for the situation where there is infiltration of fluid from the wellbore into the pore spaces of the earth formations.